A320566 Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - x^k/k!).

1, 2, 6, 23, 110, 617, 4035, 29927, 249926, 2316317, 23674841, 264329177, 3207278255, 42011308653, 591460307157, 8905905152798, 142897741683846, 2433947385964373, 43873382718719949, 834402502632550589, 16699964488044322205, 350869837371828862607, 7721899536993122262447

Offset: 0

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

• nonn

Comments

Binomial transform of A005651.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..440
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A005651, A320567, A327870.

Row sums of A327801.

Programs

Maple

seq(coeff(series(factorial(n)*exp(x)*mul((1-x^k/factorial(k))^(-1),k=1..n),x,n+1), x, n), n = 0 .. 22); # Muniru A Asiru, Oct 15 2018

Mathematica

nmax = 22; CoefficientList[Series[Exp[x] Product[1/(1 - x^k/k!), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[x + Sum[Sum[x^(j k)/(k (j!)^k), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] Total[Apply[Multinomial, IntegerPartitions[k], {1}]], {k, 0, n}], {n, 0, 22}]

Formula

E.g.f.: exp(x + Sum_{k>=1} Sum_{j>=1} x^(j*k)/(k*(j!)^k)).

a(n) = Sum_{k=0..n} binomial(n,k)*A005651(k).

a(n) ~ exp(1) * A247551 * n!. - Vaclav Kotesovec, Jul 21 2019